I don't know how much help this will be, but from my stats textbook I found in the section on estimators that:

Which proves that is a biased estimator. However, note that it is asymptotically unbiased.

Thanks, I found the algebra error in my derivation. Even so, I am still stuck on the second part. My thinking is I need to multiply by the reciprocal of the first part of the result, but I don't know what to do about getting rid of the gamma stuff.

Thanks, I found the algebra error in my derivation. Even so, I am still stuck on the second part. My thinking is I need to multiply by the reciprocal of the first part of the result, but I don't know what to do about getting rid of the gamma stuff.

Thanks for your help.

Yeah, you've got it. Multiply by the reciprocal for an unbiased estimator. The gamma stuff is just a constant as matheagle pointed out. If is even, will just be . If is odd, will be .